Edge states and the bulk-boundary correspondence in Dirac Hamiltonians

We present an analytic prescription for computing the edge dispersion $E(k)$ of a tight-binding Dirac Hamiltonian terminated at an abrupt crystalline edge. Specifically, we consider translationally invariant Dirac Hamiltonians with nearest-layer interaction. We present and prove a geometric formula that relates the existence of surface states as well as their energy dispersion to properties of the bulk Hamiltonian. We further prove the bulk-boundary correspondence between the Chern number and the chiral edge modes for quantum Hall systems within the class of Hamiltonians studied in the paper. Our results can be extended to the case of continuum theories that are quadratic in the momentum as well as other symmetry classes.

Figure 1

An illustration of Theorem 1. The gray ellipse is traced out by $\mathbf{h}(k_{\perp })={\mathbf{b}}^0+2{\mathbf{b}}^r\cos k_{\perp }+2{\mathbf{b}}^i\sin k_{\perp }$ for a fixed parallel momentum ${\mathbf{k}}_{\parallel }$ [Eq. (3)]. The dashed ellipse (${\mathbf{h}}_{\parallel }$) is $\mathbf{h}$ projected onto the plane spanned by ${\mathbf{b}}^r$ and ${\mathbf{b}}^i$. The displacement of the ellipse $\mathbf{h}$ from the dashed ellipse ${\mathbf{h}}_{\parallel }$ is given by ${\mathbf{b}}_{\perp }^0$, the component of ${\mathbf{b}}^0$ perpendicular to this plane. Theorem 1 says that an edge state exists if and only if the dashed ellipse encloses the origin (which holds true for this diagram), and its energy is determined by the displacement $\left|{\mathbf{b}}_{\perp }^0\right|$.

Figure 2

Illustration of bulk-boundary correspondence. (a) The torus traced out by $\mathbf{h}(k_x,k_y)$ for a bulk insulator with Chern number $\nu =1$. Each black loop maps out $\mathbf{h}(k_x)|{}_{k_y}$ for fixed values of $k_y$; the thick black lines guide the eye to important loops. Setting $k_y=\pi$ gives the black loop on the right that encloses the origin, meaning that there are zero-energy edge modes at this value of $k_y$. At $k_y=0$, the black loop on the left lies in the plane of the origin without containing it, indicating no edge mode at $k_y=0$. Black loops at the top and bottom ($k_v=\frac{5\pi }{3}$, $k_c=\frac{\pi }{3}$) have projections that intersect the origin, indicating the values of $k_y$, where the edge band merges with the bulk bands. (b) The band structure of the system with the edge mode shown by the thick (orange) line. The model presented here is a $p+\mathit{ip}$ superconductor described in Sec. 4b [see Eq. (17)] with parameters $t=1,{\Delta }_0=3,\mu =1$.

Figure 3

Determining the penetration depth from the ellipse ${\mathbf{h}}_{\parallel }$. The distances from the foci of the ellipse ${\mathbf{h}}_{\parallel }(k_{\perp })$ to the origin determine the characteristic decay parameter $\lambda$, which in turns gives the penetration depth $\xi =-a/2\ln \left|\lambda \right|$. In the case where ${\mathbf{h}}_{\parallel }$ traces a circle, $\left|\lambda \right|=d/r$, where $d$ is the distance of the origin to the center of the circle, and $r$ is the radius of the circle. In the general case, ${\mathbf{h}}_{\parallel }$ traces an ellipse, $\left|\lambda \right|=\frac{l+\sqrt{l^2-f^2}}{M+m}$, where $M$ and $m$ are the major and minor diameters, $f$ is the distance between the foci $\left|F_1{F}_2\right|$, and $l$ is the sum of distances $\left|{\mathit{OF}}_1\right|+\left|{\mathit{OF}}_2\right|$.